The Pierce-Birkhoff Conjecture for Curves

1992 ◽  
Vol 44 (6) ◽  
pp. 1262-1271 ◽  
Author(s):  
Murray Marshall
Keyword(s):  
Real Closed Field ◽  
Noetherian Rings ◽  
Closed Field ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Algebraic Set ◽  
Dedekind Domains ◽  
Piecewise Polynomial Functions ◽  
Relative Case

AbstractThe results obtained extend Madden’s result for Dedekind domains to more general types of 1-dimensional Noetherian rings. In particular, these results apply to piecewise polynomial functions t:C → R where R is a real closed field and C ⊆ Rn is a closed 1-dimensional semi-algebraic set, and also to the associated “relative” case where t, C are defined over some subfield K ⊆ R.

scholarly journals On the Cardinality of A Semi-Algebraic Set

1994 ◽  
Vol 1 (3) ◽  
pp. 277-286
Author(s):  
G. Khimshiashvili
Keyword(s):  
Quadratic Forms ◽  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraic Set

Abstract It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.

Spline Based Design of Cam Contours for Improved Dynamic Performance

1993 ◽  
Author(s):  
Holly K. Ault ◽  
James C. Wilkinson
Keyword(s):  
Dynamic Performance ◽  
Integrated Design ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Cam Design ◽  
Piecewise Polynomial Functions ◽  
Cam Follower ◽  
Cam Profile ◽  
Pitch Curve ◽  
Design And Manufacture

Abstract A method for the integrated design and manufacture of radial plate cams is discussed. Currently, a cam-follower system is designed by specifying constraints on the motion of the follower. The physical cam contour or cam pitch curve are not mathematically defined. The cam is manufactured from the discretized follower motion program. A new method for cam design is proposed which will produce a smooth, mathematically defined cam pitch curve while maintaining the proper constraints on the follower motion. Piecewise polynomial functions in the form of rational and/or non-rational splines may be used. Cams will be manufactured using smoothed profiles and tested for improved dynamic performance. The results of initial investigations of cam profile design for this research are presented.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

Advanced Zig-Zag Beam Theories for Sandwich Structures Analyses

2018 ◽  
Author(s):  
Matteo Filippi ◽  
Erasmo Carrera
Keyword(s):  
Sandwich Structures ◽  
Higher Order ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Displacement Fields ◽  
New Class ◽  
Dynamic Analyses ◽  
Piecewise Polynomial Functions ◽  
Carrera Unified Formulation ◽  
Beam Theories

This work aims at evaluating the capabilities of several higher-order beam formulations for stress and dynamic analyses of layered sandwich structures. The structural models are conceived within the framework of the Carrera Unified Formulation (CUF) that allows one to generate (and compare) an infinite number of displacement fields. The number and the type of functions that are selected to generate the kinematic expansions are input parameters of the problem. Besides the well-known Taylor- and Lagrange-type expansions, great attention is paid to a new class of advanced higher-order zig-zag theories, which are written as combinations of continuous piecewise polynomial functions. Numerical simulations are performed on laminated and sandwich beams with very low length-to-depth ratio values. Also, structures with soft layers made of viscoelastic materials are considered to investigate the different dissipation mechanisms.

scholarly journals On the integral representation of the endomorphisms of Mikuśinski's operator field

1971 ◽  
Vol 17 (4) ◽  
pp. 351-367 ◽  
Author(s):  
András Bleyer
Keyword(s):  
Integral Representation ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Operator Field ◽  
Piecewise Polynomial Functions

We proved in (1) that every continuous endomorphism can be generated on a subring of the field M. More precisely, the ring H of piecewise polynomial functions has the property that every isomorphism from H into M, continuous in the sequential topology of H, can be extended to a continuous endomorphism of M where the notion of continuity in M is the usual sequential one.

Sign in / Sign up

Export Citation Format

Share Document